Polynomial functions Polynomial

a polynomial function function can defined evaluating polynomial. function f of 1 argument polynomial function if satisfies.

f
(
x
)
=

a

n



x

n


+

a

n
−
1



x

n
−
1


+
⋯
+

a

2



x

2


+

a

1


x
+

a

0




{\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{2}x^{2}+a_{1}x+a_{0}}

for arguments x, n non-negative integer , a0, a1, a2, ..., constant coefficients.

for example, function f, taking real numbers real numbers, defined by

f
(
x
)
=

x

3


−
x


{\displaystyle f(x)=x^{3}-x}

is polynomial function of 1 variable. polynomial functions of multiple variables defined, using polynomials in multiple indeterminates, in

f
(
x
,
y
)
=
2

x

3


+
4

x

2


y
+
x

y

5


+

y

2


−
7.


{\displaystyle f(x,y)=2x^{3}+4x^{2}y+xy^{5}+y^{2}-7.}

an example function



f
(
x
)
=
cos
⁡
(
2
arccos
⁡
(
x
)
)


{\displaystyle f(x)=\cos(2\arccos(x))}

which, although not polynomial, polynomial function on



[
−
1
,
1
]


{\displaystyle [-1,1]}

because every



x


{\displaystyle x}





[
−
1
,
1
]


{\displaystyle [-1,1]}

true



f
(
x
)
=
2

x

2


−
1


{\displaystyle f(x)=2x^{2}-1}

(see chebyshev polynomials).

polynomial functions class of functions having many important properties. continuous, smooth, entire, computable, etc.

graphs

a polynomial function in 1 real variable can represented graph.

the graph of 0 polynomial




f(x) = 0


is x-axis.


the graph of degree 0 polynomial




f(x) = a0, a0 ≠ 0,


is horizontal line y-intercept a0


the graph of degree 1 polynomial (or linear function)




f(x) = a0 + a1x , a1 ≠ 0,


is oblique line y-intercept a0 , slope a1.


the graph of degree 2 polynomial




f(x) = a0 + a1x + a2x, a2 ≠ 0


is parabola.


the graph of degree 3 polynomial




f(x) = a0 + a1x + a2x + a3x, a3 ≠ 0


is cubic curve.


the graph of polynomial degree 2 or greater




f(x) = a0 + a1x + a2x + ... + anx , ≠ 0 , n ≥ 2


is continuous non-linear curve.

a non constant polynomial function tends infinity when variable increases indefinitely (in absolute value). if degree higher one, graph not have asymptote. has 2 parabolic branches vertical direction (one branch positive x , 1 negative x).

polynomial graphs analyzed in calculus using intercepts, slopes, concavity, , end behavior.